Hi,
I do not know what is your very first credit score when you apply for a credit card for the very first time.
A few questions for you:
*Do you already have a credit card ?
If yes, then, having been granted a GC won't affect your score ( but continue to pay your bills !

)
*Are you married ?
=> a very simple way to start a credit history is to be added to a relative's credit card. ( parent, spouse...) : indeed, if you have never had a credit card, it can be tough to get one because you are likely to be rejected for "
poor or no credit ". Also, most of stores like jewellry or underwear (

) ask for a major credit card to give you one of theirs.
So, if you are added to someone's credit card, you can " borrow" that person's score. But be careful : if you do not pay for your share, you also hurt that person's score !
* You could also apply for a secured credit card. It would be a way to build credit on your own.
You can also visit your bank and see what they have to offer.
when You Get The Green Card And You Apply For A Ss Number
By this, my guess is that you are a spouse of a USC or LPR ( if you were living legally in the US before being granted the GC, you would already have a SSN ). If true, I would recommend to try to be added on your spouse's account.
Good luck!